The robustness of nematicons, i. e. spatial solitons in nematic liquid crystals, can be exploited to implement counter-intuitive negative reflection and refraction schemes for optical signal manipulation at interfaces.
© 2007 Optical Society of America
Refraction at the interface between two transparent dielectric media is at the basis of simple steering schemes in several optical devices. Recently, the phenomenon of negative refraction (NR) in negative-index media has attracted a great deal of attention, both for realizing perfect matched lenses and for observing a reversal of the Doppler shift and Vavilov-Cerenkov radiation.[2–3] V. G. Veselago, in a pioneering theoretical work, illustrated the potential use of the so called meta-materials for NR at the interface between a normal medium -with both permittivity e and permeability μ being positive- and an abnormal left handed material -with both ε and μ being negative. In such left-handed materials the wavevector k of a plane electromagnetic wave forms a left-handed triplet with the electric E and magnetic H field vectors, i. e. the Poynting vector S is antiparallel to k. More recently, Lindell et al. pointed out that NR can occur at the interface between uniaxials exhibiting just one negative diagonal component of the dielectric tensor.
Lately, the term negative refraction has been employed more widely, including negative steering of the Poynting vector at interfaces between birefringent materials. In particular, in a series of recent articles, [6–8] researchers investigated light propagation through isotropic-uniaxial and uniaxial-uniaxial interfaces with strictly positive dielectric tensors but exhibiting both positive and negative refraction. The latter refers to incident and refracted Poynting vectors belonging to the same half-space with respect to the interface normal. Clearly, since electromagnetic left-handed triplets cannot be expected in positive index media, this is rather different from NR which occurs in left-handed media, as it has to do with the walk-off of the wave-vector k and the Poynting vector S in the propagation of an extraordinary-polarized wave. Although quite recently such “classic” NR has been explicitly re-addressed theoretically and experimentally, perhaps its first observation should be attributed to E. Bartholinus, who reported in 1669 the observation of double refraction in calcite.  In 1678 C. Huygens formalized his wave theory of light, explaining several propagation characteristics of light including double refraction (i. e., positive and negative refraction of ordinary and extraordinary waves, respectively).  For the sake of completeness in this brief historical survey, about forty years ago the Danish archaeologist T. Ramskou, with reference to ancient Icelandic sagas, suggested that the Vikings might have used double refraction of calcite to analyze the skylight polarization and find their way when navigating during cloudy days. [11–12] In 1959 M. Born and E. Wolf described a full approach to ray optics in crystals, including (although not specifically addressing) negative refraction of extraordinary (e-) waves.  In 1963 H. Kolgelnik and H. Motz showed that, for any direction of phase propagation in bulk dielectrics, the direction of the energy flux is normal to the index surface.
From the inception of second-order nonlinear optics, double refraction and walk-off have been recognized to be fundamentally important in both phase-matching processes and parametric energy exchange between waves. It has been much less emphasized the relevance and potential role of optical anisotropy in cubic nonlinear optics, particularly with reference to light self-trapping into optical spatial solitons. The latter are self-localized -hence non-diffracting- optical beams which, through a nonlinear alteration of the refractive index distribution, give rise to light-induced waveguides with large potentials in novel generations of readdressable/reconfigurable optical networks. Among the variety of materials where spatial solitons have been investigated, liquid crystals in the nematic phase have been shown to be an excellent workbench for their study at milliwatt power levels and in conjunction with a spatially non local response, i. e., one which extends well beyond the region excited by the electromagnetic field. [16–20] Nematic liquid crystals (NLC), in fact, consist of elastically-linked elongated organic molecules which can reorient under the torque generated by an electric field on the induced dipoles. Thereby, owing to the morphologic anisotropy and optical birefringence of these positive uniaxial molecules, their angular reorientation results in a refractive index increase (a positive nonlinearity) for extraordinarily-polarized waves with an electric field vector coplanar to both the wave-vector k and the major molecular axis or director n̂ . NLC reorientational nonlinearity and non locality support the generation and propagation of stable bright spatial solitons in the two transverse dimensions (2D+1), self-confined light-beams also known as Nematicons.  While NLC large birefringence and walk-off can give rise to negative refraction at an interface with glass,  voltage-controlled reorientation can also be effectively employed to steer nematicons in angle, thereby redirecting the corresponding all-optical waveguide by as much as seven degrees. Moreover, taking advantage of their intrinsic robustness and polarization conservation, nematicons can refract and even undergo total internal reflection at an interface between two NLC regions differing in index and walk-off,  resulting in various novel phenomena such as non-specular reflection in angle and power-dependent Goos-Hänchen shift. [27–28]
2. Nematicons and anomalous refraction
Let us consider an NLC cell as in Fig. 1, consisting of two parallel glass slides which confine a liquid crystal by capillarity. Defining a right-handed reference xyz, with x across the cell thickness and z along its length and perpendicular to the input interface, special treatments of the inner interfaces permit to anchor the molecules, defining a planar director (optic axis) distribution in the yz plane at an angle ρ=ρ 0 with respect to y. An external voltage can be applied through indium tin oxide electrodes deposited on the inner glass surfaces, biasing the NLC and determining the molecular reorientation in the bulk of the sample, as indicated in Fig. 1(b): the azimuth angle ξ=ξ(V) can be increased, thereby changing the refractive index experienced by a propagating e-wave at optical frequencies.
A light-beam of arbitrary polarization, impinging the input interface in z=0 with wavevector kin at an angle Φi with respect to z, is subject to double-refraction: it splits in the ordinary (o) and extraordinary (e) eigenwaves of the positive uniaxial NLC with wave vectors ko (ko<z=Φo) and ke (ke<z=Φe), respectively. Figure 2(a) displays this case at zero bias V=0 V. The Poynting vector Se associated to the e-wave lies in the extraordinary principal plane yz (שke n̂) at the walk-off angle δ with respect to ke, i. e., between ke and n̂.
In geometric optics, refraction can be determined from the wave normals. Fixing the origin, permitted eigen-wavevectors are represented by plotting k = k 0 n(r̂)r̂ , with k 0 being the vacuum wavenumber and n(r̂) the refractive index of waves propagating along r̂. If θ is the angle between n̂ and r̂, it is n(r̂) = no=n ⊥ for ordinarily polarized (“o-”) waves and n 2(r) = n 2 e(r̂) = ([cos(θ)/n ⊥]2 +[sin(θ)/n ∥]2)-1 for extraordinary (“e-”) waves. Here we use the subscripts ⊥ and ∥ with reference to the NLC molecular axis n̂ Considering all possible r̂, in each medium k describes a two-sheet surface called inverse surface of wave normals (ISWN).  The latter is a sphere (ellipsoid) for the o-waves (e-waves). In Fig. 2(b), the intersection of the ISWN with plane yz is shown for the isotropic incidence medium (dashed circle, either air or glass: for simplicity we consider just one isotropic medium of index n v) and NLC (o: solid line circle, e: solid line ellipse). Following Snell's law, the refracted wavevectors ko and ke conserve the kin projection along y, defining the angles Φo and Φe.
The Poynting vectors So and Se are perpendicular to the ISWN corresponding to the points where ko and ke contact the two sheets, respectively, with ko∥So and ke∙Se/(|ke|∙|Se|)=cos(δ), being δ = arctan(εasin2θ/(2ε ⊥ +εa + εacos2θ)) the walk-off, ε ⊥ = n 2 ⊥ and εa = n 2 ∥ - n 2 ⊥ the birefringence. Therefore, in this geometry, when δ exceeds Φe (as governed by the Snell's law) the e-wave undergoes negative refraction, as pictured in Fig. 2.
It is in line with the result above that the first observation of NR can be attributed to Bartholinus: for small angles of incidence Φi, e-wave refraction is substantially governed by the walk-off angle. Therefore, through a calcite crystal (n ⊥=1.6646, n ∥=1.4890 at λ=514nm) it is easy to observe a negative displacement of the image carried by e-rays for a range of Φi
Figure 3 shows the experimental evidence of negative refraction in the NLC configuration of Fig. 2: a mixed polarization (o and e) 1.5+1.5 mW beam from a Nd:YAG laser (λ=1064nm) is injected at a small Φi, and kiny>0 in a cell filled with commercial E7 (n ∥=1.6954 and n ⊥= 1.5038), and prepared with ρ0=60°. The image of light evolution in yz, acquired from scattered photons using a microscope and a CCD camera, displays a diffracting o-beam under normal refraction and a negatively refracted e-beam which, due to the self-focusing reorientational response, forms a spatial soliton with invariant cross-section. From the acquired images we could estimate Φo=2.7° , Φi=4.1° and δ≈5.8° which correspond to negative refraction of an e-wave.
The effect of an external bias on the NLC can be appreciated by simple geometrical considerations. Naming θ0 the angle between ke and the projection of n̂ on yz, we can write:
with δyz being the angle between ke and the projection of Se on yz and δx the S azimuthal angle. Therefore:
Expressions (4) link the Poynting vector components (Fig. 4(c)) on x and y with δ=δ(ξ) (Fig. 4(a)) and Φe=Φe(ξ) (Fig. 4(b)) as ξ increases with applied voltage, and predict negative or anomalous refraction as S is steered out of the propagation plane yz, its tip describing a semicircular line in xy.
Figure 5 shows the experimentally acquired evolution of a 3 mW nematicon as it propagates in the plane yz while the external voltage is ramped from 0 to 2V and ξ approaches 90°. As walk-off δ decreases, so does its apparent value δyz in yz, until eventually the e-beam assumes an o-ray configuration and overlaps with the (weaker) ordinary portion of the excitation, the latter obeying Snell's law.
3. Negative reflection
As wavevector conservation applies to a generic interface, the considerations in the previous section apply to the cases of total internal reflection of a beam (nematicon) traveling from an optically denser towards a rarer dielectric as well as of reflection at a metallic surface.
A straightforward configuration for the latter case is illustrated in Fig. 6, where the incident beam or spatial soliton reaches the surface of separation between NLC and a metal mirror with Poynting vector Si and wavevector ki pointing towards positive and negative y, respectively (see Fig. 6(b)). Based on the simple ISWN construction examined above, upon reflection the wavevector kR will point towards negative y and the reflected power will travel along SR with SR∙y<0 due to the small walk-off, thereby undergoing negative reflection. The experimental demonstration of this phenomenon with near-infrared light is underway.
Finally, Fig. 7 suggests the extension of negative reflection for implementing an optical resonator with an anisotropic nonlinear dielectric such as NLC. Under appropriate conditions, owing to negative reflection a self-confined wave packet (nematicon) undergoes resonance while propagating with energy flow at an angle δ with respect to the cavity mirror normals, as in Fig. 7 (b).
Although a detailed analysis is well beyond the scopes of this paper, even in the presence of significant anisotropy spatial solitons can be expected to be stable in such a resonator filled with a (spatially and temporally) nonlocal nonlinear medium.
Negative refraction as well as negative reflection can be obtained in media with strictly positive refractive index and permeability, provided there is significant birefringence and walk-off. We reported negative refraction and its voltage-tuning in uniaxial nematic liquid crystals, where a giant non local and non-resonant nonlinearity can support the propagation of self-guided beams or spatial solitons. We also pointed out a variety of voltage controlled phenomena which can be envisioned in conjunction with such all-optically induced nematicon waveguides.
The authors thank M. Kaczmarek and A. Dyadyusha (ORC, Southampton). This work was supported in part by the Italian MIUR (PRIN 2005098337).
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